QCD Wehrl and entanglement entropies in a gluon spectator model at small-$x$

This paper investigates the proton's Wehrl entropy using a gluon light-front spectator model based on soft-wall AdS/QCD wave functions, demonstrating how to decompose this entropy into entanglement and transverse components via a Husimi distribution derived from Wigner distributions, with numerical results for entanglement entropy showing agreement with CMS data.

The Gist

Imagine the proton (a tiny particle inside an atom) not as a solid marble, but as a bustling, chaotic city. Inside this city, there are tiny messengers called gluons zooming around, carrying the force that holds the city together.

For a long time, physicists have tried to measure how "disordered" or "complex" this city is. They use a concept called entropy, which is basically a measure of how much information is hidden or how chaotic a system is.

This paper is about taking a new, more detailed snapshot of this proton city to measure its entropy in a way that hasn't been done before. Here is the breakdown using simple analogies:

1. The Old Map vs. The New Map

• The Old Way (Kharzeev-Levin Model): Imagine trying to understand the traffic in a city by only counting the number of cars on a single highway. You know how many cars are there, but you don't know where they are in the city or how fast they are moving sideways. This method gives a number called Entanglement Entropy. It's a good estimate of the chaos, but it's one-dimensional. It's like knowing the total population of a city but not knowing how crowded the streets are.
• The New Way (Wehrl Entropy): The authors want to look at the whole city. They want to know not just how many cars there are, but exactly where they are and how they are moving in 3D space. To do this, they use a "map" called the Husimi distribution.

2. The "Fuzzy Camera" Problem

In the quantum world (the world of tiny particles), there is a rule called the Uncertainty Principle. It's like saying you can't take a perfectly sharp photo of a speeding car; if you focus on its speed, the location gets blurry, and if you focus on the location, the speed gets blurry.

• The Wigner Map: This is a raw, high-definition map of the proton. But because of the quantum rules, this map has weird "static" and negative numbers on it. It's like a photo with so much digital noise and glitching that you can't use it to calculate a clean number for entropy.
• The Husimi Map (The Solution): To fix the glitchy photo, the authors apply a "Gaussian smearing." Think of this as taking that noisy photo and running it through a soft-focus filter. It blurs the image just enough to get rid of the impossible negative numbers and the static, making the map smooth and positive. This smoothed-out map is the Husimi distribution.

3. Measuring the "Fuzziness" (Wehrl Entropy)

Once they have this smooth, fuzzy map, they calculate the Wehrl Entropy.

• The Analogy: If the Entanglement Entropy (the old way) is like counting the total number of people in a stadium, the Wehrl Entropy is like measuring how spread out those people are across the seats and how much they are jiggling in their seats.
• The Result: The paper finds that the Wehrl Entropy is always higher than the Entanglement Entropy. This makes sense! The old method only looked at the "longitudinal" direction (like looking down a long hallway). The new method looks at the "transverse" direction (looking at the width of the hallway too). By adding the extra dimensions of space and movement, there is more "hidden information" or "disorder" to account for.

4. The "Saturation" Filter

To make this fuzzy map work correctly, the authors had to decide how much to blur it. They used a specific "blur radius" based on something called the saturation scale.

• The Metaphor: Imagine the proton is a crowded room. If the room is empty, you can see everyone clearly. If the room is packed, people start bumping into each other, and you can't distinguish individuals anymore; they look like a single, dense blob. The "saturation scale" is the point where the room gets so full that the blur filter kicks in. The authors used a standard recipe (the GBW model) to decide exactly how much to blur the image based on how crowded the proton is.

5. What Did They Find?

The authors built a computer model of the proton using a "spectator" idea: they imagine the proton as one active gluon and a "spectator" (the rest of the proton) watching it. They tuned this model to match real-world data from particle accelerators (like the CMS experiment).

• The Big Discovery: When they compared their new "Wehrl Entropy" (the full 3D view) with the old "Entanglement Entropy" (the 1D view), they found that the new entropy is larger.
• Why it matters: The old method is like listening to a song on a mono speaker (one channel). The new method is like listening in stereo with surround sound. The new method captures the "transverse" (side-to-side) chaos that the old method missed.
• Stability: They tested their results by changing the "blur" amount and the "crowdedness" recipe. They found that their main conclusion holds up: the Wehrl entropy is a robust way to measure the proton's complexity, and it doesn't change wildly just because you tweak the settings slightly.

Summary

This paper is about upgrading the way we measure the "chaos" inside a proton.

1. Old Method: Counted the messengers (gluons) in a straight line.
2. New Method: Took a fuzzy, 3D photo of the messengers to see how they are spread out in space.
3. Result: The 3D photo reveals more chaos (entropy) than the straight-line count because it includes the sideways movement and spacing of the particles.

The authors conclude that this new "Wehrl Entropy" is a powerful tool for understanding the proton's internal structure, offering a more complete picture of the quantum world than previous methods allowed.

Technical Summary

Technical Summary: QCD Wehrl and Entanglement Entropies in a Gluon Spectator Model at Small–x

Problem Statement
Recent studies have established a connection between hadronic multiplicity in deep inelastic scattering (DIS) and entanglement entropy ($S_{EE}$), primarily through the Kharzeev–Levin (KL) model where $S_{EE} = \ln N$. However, this definition is intrinsically longitudinal, relying solely on parton distribution functions (PDFs) and failing to capture the full phase-space structure of the proton, specifically transverse degrees of freedom. Furthermore, standard quantum mechanical descriptions using Wigner distributions encounter issues in the small-$x$ regime, as these distributions are not positive-definite, preventing a direct probabilistic interpretation of entropy. There is a need for a framework that incorporates both longitudinal and transverse phase-space information while maintaining a positive-definite distribution compatible with the uncertainty principle.

Methodology
The authors employ a gluon light-front spectator model to construct a unified description of the proton's partonic structure. The methodology proceeds as follows:

1. Model Construction: The proton is modeled as a two-body Fock state consisting of an active gluon and a spin-1/2 spectator. The light-front wave functions (LFWFs) are constructed using a parametrization inspired by soft-wall AdS/QCD formalism.
2. Parameter Fixing: The free parameters of the wave function (longitudinal profile, transverse width, and normalization) are constrained by fitting the unpolarized gluon PDF to global NNPDF4.0 NNLO data at virtualities $Q = 2, 5,$ and $10$ GeV.
3. Distribution Generation:
   • Wigner Distribution: Derived from the LFWFs, providing a phase-space description in terms of the Bjorken variable $x$, impact parameter $b_\perp$, and transverse momentum $k_\perp$.
   • Husimi Distribution: Obtained by applying a minimal Gaussian smearing (Weierstrass transform) to the Wigner distribution. The smearing width is set by the inverse saturation scale, $\ell = 1/Q_s(x)$, utilizing the Golec-Biernat-Wüsthoff (GBW) model. This ensures the resulting distribution is positive-definite.
4. Entropy Calculation:
   • Entanglement Entropy ($S_{EE}$): Calculated using the KL relation $S_{EE} = \ln[xG(x)]$, derived from the fitted PDFs.
   • Wehrl Entropy ($S_W$): Calculated from the normalized Husimi distribution. The authors decompose the Wehrl entropy into the KL entanglement entropy term and a residual "transverse Wehrl entropy" ($S^\perp_W$), which accounts for transverse phase-space information.

Key Contributions

• Unified Framework: The work demonstrates how a single set of parameters, fixed by collinear PDF data, can consistently generate both longitudinal PDFs and full phase-space Wigner/Husimi distributions within a light-front spectator model.
• Decomposition of Entropy: The paper provides a formal decomposition of the Wehrl entropy, explicitly identifying the longitudinal KL entanglement entropy term and isolating a residual term ($S^\perp_W$) associated with transverse degrees of freedom (impact parameter and transverse momentum).
• Positivity and Coarse-Graining: By utilizing the Husimi distribution with a smearing scale tied to the saturation momentum, the authors circumvent the non-positivity of the Wigner distribution, enabling a well-defined semiclassical entropy calculation in the small-$x$ regime.

Results

• Validation: The model successfully reproduces the entanglement entropy extracted from CMS experimental multiplicity data across various pseudorapidity intervals and center-of-mass energies, validating the consistency of the fitted parameters.
• Virtuality Dependence:
  • The KL entanglement entropy ($S_{EE}$) increases with virtuality ($Q^2$) in the small-$x$ regime, reflecting the growth in the number of resolved gluons.
  • In contrast, the total Wehrl entropy ($S_W$) exhibits weak dependence on virtuality. This is attributed to the normalization of the Husimi distribution, which removes the overall multiplicity factor, leaving the entropy to characterize the intensive transverse phase-space structure rather than the total gluon count.
  • The transverse entropy component ($S^\perp_W$) decreases as virtuality increases.
• Hierarchy: The results consistently show $S_W > S_{EE}$. The authors interpret this as the Wehrl entropy capturing additional information loss due to the coarse-graining of transverse phase-space variables, which are absent in the purely longitudinal KL definition.
• Robustness: The qualitative behavior of the Wehrl entropy is found to be robust against variations in the Gaussian smearing width parameter ($c$) and the choice of saturation scale model (comparing GBW, rcBK, and NLO BK).

Significance and Claims
The paper claims that the Wehrl entropy offers a more complete semiclassical description of the proton's entropy content compared to the purely longitudinal entanglement entropy. While the KL model successfully describes multiplicity data, it is intrinsically limited to longitudinal momentum structure. The Wehrl entropy, by incorporating transverse degrees of freedom through the Husimi distribution, reflects the additional information loss induced by phase-space coarse-graining.

The authors posit that the Wehrl entropy is the appropriate quantity for characterizing the initial conditions of hydrodynamic evolution in heavy-ion and small-system collisions, as it encodes the degree of decoherence relevant for the emergence of a semi-classical description. The work suggests that the transverse component of the entropy ($S^\perp_W$) may be qualitatively related to the Shannon entropies of Generalized Parton Distributions (GPDs) and Transverse Momentum Distributions (TMDs). The paper concludes that these findings motivate further detailed analysis of Wehrl entropy in experimental observables, though such studies are beyond the current scope.